Question Bank • QB_EM_ADV_047 NAT • 2 Marks

Section 1: Engineering Mathematics

Vector Calculus

Subtopic: Divergence and curl

Question Statement:

The vector field $\mathbf{F} = (x + 3y)\hat{i} + (y - 2z)\hat{j} + (x + az)\hat{k}$ is solenoidal. The value of constant $a$ is ________ (answer in integer).

Answer Key:

-2

Step-by-Step Solution & Explanation

A vector field $\mathbf{F}$ is solenoidal if its divergence is identically zero: $\nabla \cdot \mathbf{F} = 0$ Compute the divergence: $\nabla \cdot \mathbf{F} = \frac{\partial}{\partial x}(x + 3y) + \frac{\partial}{\partial y}(y - 2z) + \frac{\partial}{\partial z}(x + az) = 1 + 1 + a = 2 + a$ Setting $\nabla \cdot \mathbf{F} = 0$: $2 + a = 0 \implies a = -2$
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